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Problem 2. Let F : R → R be any function (not necessarily measurable!). Prove that...

Problem 2. Let F : R → R be any function (not necessarily measurable!). Prove that the set of points x ∈ R such that

F(y) ≤ F(x) ≤ F(z)

for all y ≤ x and z ≥ x is Borel set.

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Answer #1

Borel set is any set in a topological space that can be generated from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

Now here the set(A) is where

Claim:- A is a Borel set i.e. A is generated from open sets.or closed sets.

Let a=F(y) and b=F(z), then .

F is defined on the whole real line, so it can not go to infinity at any point inside [y,z], so   is a union of some open and closed sets, which proves our claim that A is generated from open sets.or closed sets, so it Borel.

Thanks

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